If the same channel is displayed on all three, they are showing the exact same digital stream. The Cable Box and TivoHD will map a different number to the channel. QAM direct will display (most likely) the physical channel or the stations broadcast number while the other two will display Cables virtual number which will match the cable guide number. (the QAM won't)

The Box and TivoHD could display a poorer picture due to the connections used and the internal conversion needed to match the desired output connector. (HD Component, which is analog, or DVI/HDMI)

So, in theory, the internal QAM tuner should always display the better picture ....... however, internal processing in the TV can do weird things depending on the set. On one TV, I have seen that all of the inputs, including the DVI, were converted to YPbPr before passing to the output scalar, which converted them again. If you have a TV with poor processing circuits, then the DVI/HDMI (but not Component) could look better because it bypasses the analog circuitry. The QAM or OTA internal stream should never see analog anywhere in the set but a cheap company could have purchased a circuit design that requires analog as an input/output at some point. You'd have to see a detailed block diagram of the signal path inside the set to be certain. (and someone who understood what the different blocks did)

When I started writing this, I was sure the internal QAM would always be = or better, and then I remembered that there are Korean and Chinese designs. So, my final answer is that the internal QAM will always be better or equal to a Component (Red/Blue/Green) input (and never worse) and should be better or equal to DVI/HDMI, but might not if you have a poorly designed set.

Most people that see noticeable differences usually have not calibrated their set for each input (one setting does not control all inputs on most newer sets, each one will have it's own preferences). The internal tuner will usually have more setup options than the external inputs ... or they are accidentally displaying the SD QAM stream rather than the HD QAM stream for a given channel since they tend to both be present, often with confusing channel numbers.

As long as the box isn't a Moto which can only output 1 res so you double scale the other res. Moto&SA HD STBs haven't been updated in a long time (not counting the cablecard update) so their scalars are probably crap compared to any TV today.